Consider knowledge graphs as a motivating example. The same entity can exhibit multiple local logical patterns given by hierarchical, symmetric and anti-symmetric relations with other entities [1]}. A similar case can be found in WordNet [2]}. Words inside a synset show a full-mesh layout with synonyms, and tree-like hypernym relations with other synsets. As illustrated in Figure REF , Cartesian products of trees and grids mix the tree- and grid-like features globally. On the other hand, rooted products of trees with grids and of grids with trees mix these features at different levels and scales.
Thus, these graphs reflect to a greater extent the complexity of intertwining and varying structure in different regions, making them a better approximation of real-world datasets. Here we consider the rooted product Tree \(\diamond \)  Grids of a tree and 2D grids as well as the rooted product Grid \(\diamond \)  Trees of a 2D grid and trees.
Iterating the procedure of rooted products, taking for example the rooted products with different graphs, or a rooted product of a rooted product, one can generate graphs with diverse structures at varying scales.
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